Surplus Free Poles of Approximating Rational Functions

Surplus Free Poles of Approximating Rational FunctionsAuthor(s): J. L. WalshSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 52, No. 4 (Oct. 15, 1964), pp. 896-901Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/72382 .

Accessed: 07/05/2014 19:18

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

National Academy of Sciences is collaborating with JSTOR to digitize, preserve and extend access toProceedings of the National Academy of Sciences of the United States of America.

http://www.jstor.org

This content downloaded from 169.229.32.136 on Wed, 7 May 2014 19:18:17 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=nas

http://www.jstor.org/stable/72382?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


896 MATHEMATICS: J. L. WALSH PRoc. N. A. S.

The solvolysis rate constants of IV and the undeuterated anmalogue were ineasured in 50 per cent ethanol at 50?. The results were kH = 8.30-10 -4sec-' anld kD =

7.58-10-4 sec-l (kH/kD = 1.09). In Table 2 the observed isotope effect is comil-

pared to those reported for solvolyses of some other tertiary halides. It can be seen that in our case the rate retardation caused by three 3-deuteriums (9%) is indeed much smaller than generally observed (40-50%).

Work is in progress to test the scope of the latter observation and the validity of the interpretation.

* Taken in part from the Ph.D. thesis of M. N., University of Zagreb, 1963. t Present address: Gates and Crellin Laboratories, California Institute of Technology,

Pasadena. 1 Brown, H. C., in The Transition State, Special Publ. No. 16, Chemical Society, London, 1962,

pp. 140-158, 174-178; Brown, H. C., and F. J. Chloupek, J. Am. Chem. Soc., 85, 2322 (1963); Brown, H. C., and H. M. Bell, J. Am. Chem. Soc., 85, 2324 (1963); Winstein, S., A. H. Lewin, and K. C. Pande, J. Am. Chem. Soc., 85, 2324 (1963); Brown, H. C., F. J. Chloupek, and Min-Hon Rei, J. Am. Chem. Soc., 86, 1247, 1248 (1964). See also refs. 6 and 7.

2 Cox, E. F., M. C. Caserio, M. S. Silver, and J. D. Roberts, J. Am. Chem. Soc., 83, 2719 (1961), and previous papers.

3 Wilt, J. W., and D. D. Roberts, J. Org. Chem., 27, 3430 (1962). 4 Sneen, R. A., K. M. Lewandowsky, I. A. I. Taha, and B. R. Smith, J. Am. Chem. Soc., 83,

4843 (1961). 5 Breslow, R., J. Lockhart, and A. Small, J. Am. Chem. Soc., 84, 2793 (1962). 6 Corey, E. J., and Hisashi Uda, J. Am. Chem. Soc., 85, 1788 (1963). 7 Brown, H. C., F. J. Chloupek, and Min-Hon Rei, J. Am. Chem. Soc., 86, 1246 (1964). 8 Borcic, S., M. Nikoletic, and D. E. Sunko, J. Am. Chem. Soc., 84, 1615 (1962). 9 Nikoletic, M., S. Borcic, and D. E. Sunko, Abstracts, XIXth International Congress of Pure

and Applied Chemistry, London, 1963, sect. A, p. 26. 10 In a recent article D. D. Roberts [J. Org. Chem., 29, 294 (1964)] found only a fourfold rate

acceleration in acetolysis of (1-methylcyclopropyl)-carbinyl tosylate relative to cyclopropylcarbinyl tosylate, in agreement with our results.

11 Halevi, E. A., in Progress in Physical Organic Chemistry (New York: John Wiley and Sons, 1963), vol. 1, p. 189.

12 Shiner, V. J., Jr., and J. S. Humphrey, Jr., J. Am. Chem. Soc., 85, 2416 (1963). 13 Shiner, V. J., Jr., and J. G. Jewett, J. Am. Chem. Soc., 86, 945 (1964).

SURPLUS FREE POLES OF APPROXIMATING RATIONAL FUNCTIONS*

BY J. L. WALSH

DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY

Communicated August 24, 1964

Under broad conditions, as expressed in Theorem 1 below, rational functions which approximate on a point set E of the z plane to a function analytic on E and

meromorphic in a region containing E have free poles which approach the respective poles of f(z). However, if the rational functions have an excess of free poles, the

question arises as to the asymptotic behavior of those poles, and the possible effect on degree of convergence on E and elsewhere if those poles are suppressed.

The present note makes a modest contribution to this study, in both general theory and by specific examples.

896 MATHEMATICS: J. L. WALSH PRoc. N. A. S.

The solvolysis rate constants of IV and the undeuterated anmalogue were ineasured in 50 per cent ethanol at 50?. The results were kH = 8.30-10 -4sec-' anld kD =

7.58-10-4 sec-l (kH/kD = 1.09). In Table 2 the observed isotope effect is comil-

pared to those reported for solvolyses of some other tertiary halides. It can be seen that in our case the rate retardation caused by three 3-deuteriums (9%) is indeed much smaller than generally observed (40-50%).

Work is in progress to test the scope of the latter observation and the validity of the interpretation.

* Taken in part from the Ph.D. thesis of M. N., University of Zagreb, 1963. t Present address: Gates and Crellin Laboratories, California Institute of Technology,

Pasadena. 1 Brown, H. C., in The Transition State, Special Publ. No. 16, Chemical Society, London, 1962,

pp. 140-158, 174-178; Brown, H. C., and F. J. Chloupek, J. Am. Chem. Soc., 85, 2322 (1963); Brown, H. C., and H. M. Bell, J. Am. Chem. Soc., 85, 2324 (1963); Winstein, S., A. H. Lewin, and K. C. Pande, J. Am. Chem. Soc., 85, 2324 (1963); Brown, H. C., F. J. Chloupek, and Min-Hon Rei, J. Am. Chem. Soc., 86, 1247, 1248 (1964). See also refs. 6 and 7.

2 Cox, E. F., M. C. Caserio, M. S. Silver, and J. D. Roberts, J. Am. Chem. Soc., 83, 2719 (1961), and previous papers.

3 Wilt, J. W., and D. D. Roberts, J. Org. Chem., 27, 3430 (1962). 4 Sneen, R. A., K. M. Lewandowsky, I. A. I. Taha, and B. R. Smith, J. Am. Chem. Soc., 83,

4843 (1961). 5 Breslow, R., J. Lockhart, and A. Small, J. Am. Chem. Soc., 84, 2793 (1962). 6 Corey, E. J., and Hisashi Uda, J. Am. Chem. Soc., 85, 1788 (1963). 7 Brown, H. C., F. J. Chloupek, and Min-Hon Rei, J. Am. Chem. Soc., 86, 1246 (1964). 8 Borcic, S., M. Nikoletic, and D. E. Sunko, J. Am. Chem. Soc., 84, 1615 (1962). 9 Nikoletic, M., S. Borcic, and D. E. Sunko, Abstracts, XIXth International Congress of Pure

and Applied Chemistry, London, 1963, sect. A, p. 26. 10 In a recent article D. D. Roberts [J. Org. Chem., 29, 294 (1964)] found only a fourfold rate

acceleration in acetolysis of (1-methylcyclopropyl)-carbinyl tosylate relative to cyclopropylcarbinyl tosylate, in agreement with our results.

11 Halevi, E. A., in Progress in Physical Organic Chemistry (New York: John Wiley and Sons, 1963), vol. 1, p. 189.

12 Shiner, V. J., Jr., and J. S. Humphrey, Jr., J. Am. Chem. Soc., 85, 2416 (1963). 13 Shiner, V. J., Jr., and J. G. Jewett, J. Am. Chem. Soc., 86, 945 (1964).

SURPLUS FREE POLES OF APPROXIMATING RATIONAL FUNCTIONS*

BY J. L. WALSH

DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY

Communicated August 24, 1964

Under broad conditions, as expressed in Theorem 1 below, rational functions which approximate on a point set E of the z plane to a function analytic on E and

meromorphic in a region containing E have free poles which approach the respective poles of f(z). However, if the rational functions have an excess of free poles, the

question arises as to the asymptotic behavior of those poles, and the possible effect on degree of convergence on E and elsewhere if those poles are suppressed.

The present note makes a modest contribution to this study, in both general theory and by specific examples.



VOL. 52, 1964 MATHEMATICS: J. L. WALSH 897

To be more explicit, we envisage a point set E which is closed and bounded, whose complement K is regular in the sense that it possesses Green's function G(z) with pole at infinity. We denote generically by F, the locus G(z) = log O- (>0) in K, and by E,. the (open) interior of F,; we set 4m(z) =-- exp (G(z)).

A rational function of the form

aozj + alzi-l + ... +aj bozk + bl k-- + ... +bk

where the denominator does not vanish identically is said to be of type (j,k). Then we havel, 2

THEOREM 1. Let E be a closed bounded set whose complement is connected and regular, let f(z) be analytic on E, and let p ( a o ) be the largest number such that f(z) is meromorphic with precisely v(> O) poles in Ep. Suppose the rational functions Rnv(z) of respective types (n,v) satisfy for the Tchebycheff (uniform) norm on E

limn sup \lf(z) - Rn ()Z)ll'/" < l/p. (2) n-- co

Let D denote Ep with the v poles of f(z) deleted. Then for n sufficiently large the function R,n(z) has precisely v finite poles, which approach respectively the v poles of f(z) in Ep. The functions Rn,(z) converge to f(z) throughout D. For any closed bounded continuum S in D and in the closed interior of E,, 1 < o- < p, we have

lirm sup [max ff(z) - Rna(z) , z on S]i n a c/p. (3) n - ao

Hitherto the sequence R,,(z) need not be defined for every n, but below the Rna(z) shall be so defined.

The equality sign holds in (2), and in (3) if S contains a point of r,; moreover, (2) is valid for the functions Rny(z) of types (n,v) of best approximation to f(z) on E.

The primary topic of the present note is indicated in THEOREM 2. Let f(z) in Theorem 1 have either a singularity other than a pole or at

least J - v + 1 poles on Fp, A > v. Let Rn,(z) be a sequence of rational functions of

type (n,,) not necessarily defined for every n, satisfying

limr sup If/(z) - Rn,(z)llll ' l/p. (4) n --- co

Let each Rn,,(z) have precisely one simple pole at z = ac approaching the point a in

Ep - E, where f(z) is analytic at z = a. If Rn(z) denotes the principal part of the pole of Rn,,(z) at ,n, we have

lim sup [max IRn(z) |, z on Eo]l" n (O(a)/p, (5) n -- co

lim sup 1If(z) - [Rni(z) - Rn(z)]ll" _ < ((a)/p, (6) n -o

where Eo is any closed bounded set not containing a. If the finite poles of the R,, are not bounded in their totality, Theorem 2 can ob-

viously be applied to any subsequence of the R,,(z) whose finite poles are bounded in their totality, and canr also be applied to any subsequence of the Rn,(z), some of whose poles (j (say, X in number) become infinite. In the latter case we simply re- place each Rn,,(z) of the subsequence by Rn,(z) multiplied by the factor



898 MATHEMATICS: J. L. WAL.SH PROC. N. A. S.

Z - Oj

II X' <X, j= 1 -_j

whenever If I exceeds a number B, where both Ep and all the , - X poles of R,,(z) which are bounded lie in z I < B; this replacement does not alter the hypothesis (4) nor the conclusion (5) and (6).

We assume henceforth that the finite poles of the R,,(z) are bounded in their totality.

It is a consequence of the method of proof2 of Theorem I that each pole of f(z) in Ep is the limit of poles of R,,(z) of at least the same total multiplicity, so at most iu - v finite poles of the original R,,n(z) can become infinite.

A chief difficulty in the proof of Theorem 2 is that the poles of the Rn,(z) may lie everywhere dense in a subregion of Ep - E, or on a curve in Ep - E (compare ref. 2, Theorem 5). The function which is the limit of the sequence Rn,,(z) might con- ceivably then not be a monogenic analytic function even in two disjoint regions of convergence not separated by rp. This eventuality cannot arise in the situation of Theorem 1, where the v finite poles of the R,,,(z) approach the v poles of f(z) in

Ep- E, so the limit points of the poles of the R,,(z) cannot separate the plane. Given a neighborhood of a of Theorem 2 containing no pole of f(z) nor point of

Eo, and lying in some E,, a < p, we choose in that neighborhood a circle y with center a, containing on or within it no limit point of the poles other than a of the

Rn,,(z). By use of the corresponding function of Theorem 1 we have from (2) and (4)

lim sup \lRn,(z) - Rn,(z)ll'n < l/p. (7) n ---' co

Consequently, by slight modification of Lemma 3 of reference 1, we have

lim sup [max IR,,(z) - Rn,,(z) ,z on y]ll" < a/p. (8) n co

By (3), we may write from (8)

lim sup [max |f(z) - Rn,,(z) , on -yll] < c/p. (9) n --> co

We have for z on Eo for n sufficiently large

-Rn) 1 Zy f(t) - Rn.(t) dt. (10)

27r1 i 7 t- z

By (9) we deduce (5) with the second member replaced by c/p, and letting ca approach 4(a) establishes (5). Inequality (6) follows from (4), and (5) with Eo = E.

It may seem surprising that the second member of (5) is not I/p, for Rn(z) ap- proaches zero and is merely of the form Bn/(z - an) where ca -- a. Theorem 4 of reference 1 asserts that such a function has the same degree of convergence (n --- oo) on every closed point set not containing a. However, there is in our hypothesis no presumption that Rn,(z) - R,,(z) has no pole at infinity; such a pole of order n must be considered a possibility. Here a counterexample is useful. Let E be Iz I < 1/p (<1), rp be z| = I, let f(z) satisfy the hypothesis of Theorem 2, and set

u = + 1.



VoIt. 52, 1964 MATHEMATICS: J. L. WALSH 899

zn Rn,+p,,( Rn) p(z)+ -- , (11)

where Rn,(z) is the function of Theorem 1, chosen to exist for every n. We may choose the ,n in many ways; for the moment choose n~ -- a in Ep - E. If the equality sign holds in (2), it also holds in (4), (5), and (6), with @(z) p |z |; we have Rn(z) = an/ (z - a,).

Theorem 2 can be improved in the case u = v + 1, where we suppose (4) with

R+,,a, (z) - Rn,(Z) + B,, (12)

where Rn,(z) satisfies (2). Indeed (5) with 4a(a) replaced by unity follows at once [i.e., without (10)] for z on E, hence follows [ref. 1, Theorem 4] for Eo any closed set of the plane not containing c. Inequality (6) is here weaker than the inequality (2).

If (12) is modified so that precisely k poles of R,n+, ,,(z) approach a, k < - v, the same method shows that (5) with f((a) replaced by unity is valid for z on E, hence follows as before for Eo any closed set of the plane not containing a. More- over, precisely k zeros of Rn+,, ,(z) also approach a; compare Lemma I of reference 1.

The example (11) is also of interest if the an are everywhere dense in Ep - E or on E,, or _ p; compare here [ref. 2, Theorem 5].

The proof of Theorem 2 with minor changes yields: COROLLARY 1. Theorem 2 remains valid if the hypothesis is modified so that poles

of R,,(z) of total multiplicity X approach a point a in Ep - E which is either a point of analyticity of f(z) or a pole of f(z), necessarily of multiplicity not greater than X. Here R,(z) denotes the sum of the principal parts of the totality of poles of R,,n(z) approaching a less the principal part of the pole (if any) off(z) at a.

It may be noted that any subsequence of the R,,n(z) whose poles have a limit point in Ep - E admits a subsequence satisfying the conditions of Theorem 2 or of Corollary 1. Iteration of this remark yields a subsequence of the R,,n(z) having no more than A limit points of poles in Ep - E, including the v poles of f(z) in Ep - E.

COROLLARY 2. Theorem 2 remains valid if the hypothesis is modified so that the limit points of some or all of the extraneous poles of Rn,,(z) in Ep - E (that is, poles in addition to the v poles approaching the v poles of f(z) in Ep) lie interior to a closed region E' in Ep where E' contains no other limit points of poles of Rn,,(z) except perhaps poles of f(z). Precise analogues of (5) and (6) are valid, where I(a) = [max a(z), z in E'], and Rn(z) is the sum of the principal parts of the totality of poles of R,n(z) approaching points of E' less the principal parts of the poles of f(z) in E'.

Corollary 2 applies even if E' is a closed multiply connected region, and contains points of E in a lacuna. Corollary 2 follows at once, except in the case (included below) that E' is a subset of E.

COROLLARY 3. Let the hypothesis of Theorem 2 be modified so that some or all of the

poles of R,,(z) in Ep approach E, and suppose 3 in Ep - E can be chosen so that no poles of f(z) lie interior to rP(),, and no limit points of poles of Rn,,(z) lie there except on E. Let Rn(z) denote the sum of the principal parts of the poles of R,,(z) interior to Pa(--. Then (5) and (6) are valid with 4((a) replaced by unity.



900 MAT 'HEMATICS: J. L. WALSH PROC. N. A. S.

The proof of Theorem 2 is essentially sufficiet to establish Corollary 3. Choose y as F,, 1 < T < (). We use (10) for z on Fr(9), and deduce

lim sup [max R,n(z) , z on r(rB)]l/n T7/p, n o--

whence by the method of proof of (9)

lim sup [max lf(z) - [Rn,,() - Rn,(z) , z on r.'(]'/) ] n < )/p. n -- oo

The latter inequality yields (IP(0) -- 1), the assertion regarding (6), and the assertion regarding (5) follows by (4).

COROLLARY 4. In Theorem 2 let the condition on the poles of the Rn,,(z) be deleted. Then on any closed set Eo in Ep containing no limit points of poles of the R,,(z) and no pole of f(z) we have

lim sup [max If(z) - Rn,(z) , z on. Eo]""n ? a(c)/p, (13)

n o-- C

where ((ca) = [max (P(z), z on Eo]. The proof of (13) follows precisely the proof of (9). The corollaries are clearly related to

THEOREM 3. In Theorem 2 let the R,,(z) be defined for every n, and modify the

hypothesis so that all the limit points of the poles of the R,,(z) in Ep + p lie on the

closed subset E' of Ep. Then the equality sign holds in (4). Suppose the second member of (4) is l/pl(< l/p); we shall reach a contradiction.

We have

lim sup jRRn,+, j(z) - Rn,,(z)/l

" 1/pI, n -a- co

and by a slight modification of Lemma 3 of reference 1,

lim sup [max [Rn+l, (z) - R,,(z) , z on rl] < a/pl, (14) n oo --

where Pr, < p < pi, contains E' in its interior. Further use of essentially that

lemma, with (14) as hypothesis, gives

lim sup [max Rn+l, ,(z) - Rn,(z) , z on ,l]"/ < rT/pl, n a)* co

by virtue of the equation [r],]/ = r,; here we choose p < T < pi, and also choose

T so that no limit points of poles of the Rn,(z) lie in the closed annular regions be-

tween and bounded by Pr and P,. Consequently the sequence R,,(z) converges

uniformly on both PF and F,, has no limit point of poles in the set of annular regions bounded by them, and converges uniformly to some analytic function F(z) in those

annular regions. Corollary 4 implies F(z) == f(z) in the parts of those annular

regions in Ep, hence also throughout those regions, contrary to the definition of p in Theorem 2.

Under the conditions of Theorems 1 and 2, the class Rn,,() includes the class

Rn,(z), so for the respective functions of best approximation to f(z) on E we have

lf(z) - Rn,,,a(z)l ilf(Z) - Rnv(Z)\i

Nevertheless, for the functions of best approximation the first member of (2) is



VOL. 52, 1964 MATHEMATICS: R. E. MILES 901

l/p, and under the conditions of Theorem 3 the first member of (4) is also I/p. Theorem 2 and later results including the corollaries have application to the dual

problem, approximation by functions of type Rn,(z) considered in reference 3. The methods of the present note for the study of convergence and degree of con-

vergence on subsets of Ep apply when the number of free poles of the approximating: rational functions is finite, and by dualization apply when the number of their free: zeros is finite; they do not apply when the number of both free poles and zeros is infinite.

* This research was sponsored (in part) by the U.S. Air Force Office of Scientific Research. 1 Walsh, J. L., Math. Ann., 155, 252-264 (1964). 2 Walsh, J. L., "The convergence of sequences of rational functions of best approximation, II,"

in preparation. 3 Walsh, J. L., these PROCEEDINGS, 50, 791-794 (1963).

RANDOM POLYGONS DETERMINED BY RANDOM LINES IN A PLANE*

BY R. E. MILES

EMMANUEL COLLEGE, CAMBRIDGE, ENGLAND

Communicated by S. A. Goudsmit, July 22, 1964

Introduction. Consider a system of straight lines in different directions, dis- tributed at random homogeneously throughout a plane (see Fig. 1). This paper presents the main results of a study concerned primarily with the statistical properties of the aggregates of polygons into which such systems divide the plane. This problem was first considered by Goudsmit,2 who was able to derive a number of properties. The present study utilizes more powerful and more general methods, and consequently yields many additional results. In order to give a brief and sim- plified description of the theory comprehensible to nonspecialists interested in appli- cations, a number of mathematical points must needs be glossed over or even ignored; the following treatment should therefore only be regarded as heuristic. Parentheses will often signify heuristic ideas. Only a knowledge of elementary probability theory is presupposed. The precise specifications of the system are followed by the sequence of main results. The second and concluding part of this paper, which will appear in the November issue of these PROCEEDINGS, contains a short account of methods and of an application.

The Line System ?.--By way of notation, the equation of any line in the (x,y) plane may be written as

p = x cos 0 + y sin 0 (- < p < , 0 < 0 < Tr),

where p is the signed (i.e., positive or negative) length of the perpendicular, or its

distance, from the origin 0 and 0 is its orientation-the angle this perpendicular makes with Ox. There is thus a one-to-one correspondence between lines in the

(x,y) plane and points in the strip 0 m 0 < 7r of the (p,O) plane. The distances ... p-2 ~ p-l I po P pl < p2 $ . .. of the lines from 0 constitute

the coordinates of the events of a Poisson (or purely random) process on a one-dimen-

VOL. 52, 1964 MATHEMATICS: R. E. MILES 901

l/p, and under the conditions of Theorem 3 the first member of (4) is also I/p. Theorem 2 and later results including the corollaries have application to the dual

problem, approximation by functions of type Rn,(z) considered in reference 3. The methods of the present note for the study of convergence and degree of con-

vergence on subsets of Ep apply when the number of free poles of the approximating: rational functions is finite, and by dualization apply when the number of their free: zeros is finite; they do not apply when the number of both free poles and zeros is infinite.

* This research was sponsored (in part) by the U.S. Air Force Office of Scientific Research. 1 Walsh, J. L., Math. Ann., 155, 252-264 (1964). 2 Walsh, J. L., "The convergence of sequences of rational functions of best approximation, II,"

in preparation. 3 Walsh, J. L., these PROCEEDINGS, 50, 791-794 (1963).

RANDOM POLYGONS DETERMINED BY RANDOM LINES IN A PLANE*

BY R. E. MILES

EMMANUEL COLLEGE, CAMBRIDGE, ENGLAND

Communicated by S. A. Goudsmit, July 22, 1964

Introduction. Consider a system of straight lines in different directions, dis- tributed at random homogeneously throughout a plane (see Fig. 1). This paper presents the main results of a study concerned primarily with the statistical properties of the aggregates of polygons into which such systems divide the plane. This problem was first considered by Goudsmit,2 who was able to derive a number of properties. The present study utilizes more powerful and more general methods, and consequently yields many additional results. In order to give a brief and sim- plified description of the theory comprehensible to nonspecialists interested in appli- cations, a number of mathematical points must needs be glossed over or even ignored; the following treatment should therefore only be regarded as heuristic. Parentheses will often signify heuristic ideas. Only a knowledge of elementary probability theory is presupposed. The precise specifications of the system are followed by the sequence of main results. The second and concluding part of this paper, which will appear in the November issue of these PROCEEDINGS, contains a short account of methods and of an application.

The Line System ?.--By way of notation, the equation of any line in the (x,y) plane may be written as

p = x cos 0 + y sin 0 (- < p < , 0 < 0 < Tr),

where p is the signed (i.e., positive or negative) length of the perpendicular, or its

distance, from the origin 0 and 0 is its orientation-the angle this perpendicular makes with Ox. There is thus a one-to-one correspondence between lines in the

(x,y) plane and points in the strip 0 m 0 < 7r of the (p,O) plane. The distances ... p-2 ~ p-l I po P pl < p2 $ . .. of the lines from 0 constitute

the coordinates of the events of a Poisson (or purely random) process on a one-dimen-



Documents

Surplus Free Poles of Approximating Rational Functions